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Full text of "Treatise On Analysis Vol-Ii"

10   CONVOLUTION OF TWO FUNCTIONS        291

function (5-, x)^g(x'1s)f(x) = g(5'1x)f(x) is (JJL  0)-integrable, and the
right-hand side of (14.10.9.1) Is then equal to

(14.10.9.2)

Similarly, condition (b) means that the function (s9 x)t-+g(x)f(sx) is 0
integrable, and the left-hand side of (14.10.9.1) is then equal to

(14.1 0.9.3)                        II g(x)f(sx) dn(s) df(x).

11

But from the theorem of Lebesgue-Fubini and the left-invariance of /?, it
follows that one of the integrals (14.10.9.2), (14.10.9.3) exists if and only if
the other one exists, and then both are equal.

PROBLEMS

1.    (a)   Let g be a j3-measurable function of compact support on G. Show that / and
g are convolvable and that/*# is continuous on G in the following two cases:

(i)   g is essentially bounded and / is locally /3-integrable;
(ii)   g e J^(G, ]8), /is jS-measurable and/2 is locally jS-integrable.
(b)   Assume that G is unimodular. Show that if /e ^(G, ]8) and g e J*?J(G, ]8),

where p^\9q^\ and - 4- - ^ 1, then / and g are convolvable, f*ge J2?c(G, ]8)

p   %

where - = - -f - 1, and Nr(/* g)  : Na(/)N/#) (W. Young's inequality). (Consider
r     p     q

first the case where - + - = 1, then use (14,10.6.1) and the Riesz-Thorin theorem

P     Q
(Section 13.17, Problem?).)

2.    Let G be a compact group, ]8 the Haar measure on G for which j8(G) == 1, and let
A, B be two /3-integrable sets. Show that for each e > 0 there exists x e G such that
)3(A n xB) g (1 -f c))3(A)]5(B). Deduce that, for each integer H such that ]8(A) ^ l//i,
there   exist 71 points xt, ..,, xn in G such that jS(xiA u x2 A u    u xfj A) ^ ^.
(Consider the sets .*/  CA.)

3.    Let G be a compact commutative group, ]8 the Haar measure on G for which /3(G) = 1,
and g a /3-measurable function on G. Let (rn) be the orthonormal system of Rademacher
functions (Section 13.21, Problem 10). Let a > 0 and let A be the set of x e G such
that |000| >oc.

Let n be an integer J> 1 such that n  /3(A) J> 1. Show that there exists a j3-measur-
able set B such that j3(B) > |, and  points si,...,sninG such that, if we put F(x, t) 

n

V rk(t)g(skx\ the following is true: for each x e B, there exists a finite union 100 of
fc=i
intervals in [0,1] such that (i) A(I(x)) ^ J (A being Lebesgue measure); (ii) for allh the